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DERIVATIVES OF THETA FUNCTIONS AS TRACES OF PARTITION EISENSTEIN SERIES

Part of: Combinatorics Discontinuous groups and automorphic forms Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  20 January 2025

TEWODROS AMDEBERHAN Open the ORCID record for TEWODROS AMDEBERHAN [Opens in a new window] ,
KEN ONO Open the ORCID record for KEN ONO [Opens in a new window]  and
AJIT SINGH Open the ORCID record for AJIT SINGH [Opens in a new window]
TEWODROS AMDEBERHAN
Affiliation:
Department of Mathematics, Tulane University New Orleans LA 70118 USA tamdeber@tulane.edu
KEN ONO*
Affiliation:
Department of Mathematics, University of Virginia Charlottesville VA 22904 USA
AJIT SINGH
Affiliation:
Department of Mathematics, University of Virginia Charlottesville VA 22904 USA ajit18@iitg.ac.in
*
ko5wk@virginia.edu
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Abstract

In his “lost notebook,” Ramanujan used iterated derivatives of two theta functions to define sequences of q-series $\{U_{2t}(q)\}$ and $\{V_{2t}(q)\}$ that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of “partition Eisenstein series,” extensions of the classical Eisenstein series $E_{2k}(q),$ defined by

$$ \begin{align*}\lambda=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \longmapsto \ \ \ E_{\lambda}(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}. \end{align*} $$

For functions $\phi : \mathcal {P}\mapsto {\mathbb C}$ on partitions, the weight $2n$ partition Eisenstein trace is

$$ \begin{align*}\operatorname{\mathrm{Tr}}_n(\phi;q):=\sum_{\lambda \vdash n} \phi(\lambda)E_{\lambda}(q). \end{align*} $$

For all t, we prove that $U_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _U;q)$ and $V_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _V;q),$ where $\phi _U$ and $\phi _V$ are natural partition weights, giving the first explicit quasimodular formulas for these series.

Keywords

partition Eisenstein seriesquasimodular forms

MSC classification

Primary: 11F03: Modular and automorphic functions 05A17: Partitions of integers
Secondary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 12
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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